Translate an Equation and Solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign. Table 8.9 reminds us of some of the words that translate to the equal sign.

Table 8.9

Equals (=)

is

is equal to

is the same as

the result is

gives

was

will be

Let’s review the steps we used to translate a sentence into an equation.

HOW TO: TRANSLATE A WORD SENTENCE TO AN ALGEBRAIC EQUATION

Step 1. Locate the "equals" word(s). Translate to an equal sign.

Step 2. Translate the words to the left of the "equals" word(s) into an algebraic expression.

Step 3. Translate the words to the right of the "equals" word(s) into an algebraic expression.

Translate and Solve Applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

Example 8.11:

The Robles family has two dogs, Buster and Chandler. Together, they weigh 71 pounds. Chandler weighs 28 pounds. How much does Buster weigh?

Solution

Read the problem carefully.

Identify what you are asked to find, and choose a variable to represent it.

Practice Makes Perfect

Solve Equations Using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given value is a solution to the equation.

Is y = \(\frac{1}{3}\) a solution of 4y + 2 = 10y?

Is x = \(\frac{3}{4}\) a solution of 5x + 3 = 9x ?

Is u = \(− \frac{1}{2}\) a solution of 8u − 1 = 6u?

Is v = \(− \frac{1}{3}\) a solution of 9v − 2 = 3v?

In the following exercises, solve each equation.

x + 7 = 12

y + 5 = −6

b + \(\frac{1}{4}\) = \(\frac{3}{4}\)

a + \(\frac{2}{5}\) = \(\frac{4}{5}\)

p + 2.4 = −9.3

m + 7.9 = 11.6

a − 3 = 7

m − 8 = −20

x − \(\frac{1}{3}\) = 2

x − \(\frac{1}{5}\) = 4

y − 3.8 = 10

y − 7.2 = 5

x − 15 = −42

z + 5.2 = −8.5

q + \(\frac{3}{4}\) = \(\frac{1}{2}\)

p − \(\frac{2}{5}\) = \(\frac{2}{3}\)

y − \(\frac{3}{4}\) = \(\frac{3}{5}\)

Solve Equations that Need to be Simplified

In the following exercises, solve each equation.

c + 3 − 10 = 18

m + 6 − 8 = 15

9x + 5 − 8x + 14 = 20

6x + 8 − 5x + 16 = 32

−6x − 11 + 7x − 5 = −16

−8n − 17 + 9n − 4 = −41

3(y − 5) − 2y = −7

4(y − 2) − 3y = −6

8(u + 1.5) − 7u = 4.9

5(w + 2.2) − 4w = 9.3

−5(y − 2) + 6y = −7 + 4

−8(x − 1) + 9x = −3 + 9

3(5n − 1) − 14n + 9 = 1 − 2

2(8m + 3) − 15m − 4 = 3 − 5

−(j + 2) + 2j − 1 = 5

−(k + 7) + 2k + 8 = 7

6a − 5(a − 2) + 9 = −11

8c − 7(c − 3) + 4 = −16

8(4x + 5) − 5(6x) − x = 53

6(9y − 1) − 10(5y) − 3y = 22

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

Five more than x is equal to 21.

The sum of x and −5 is 33.

Ten less than m is −14.

Three less than y is −19.

The sum of y and −3 is 40.

Eight more than p is equal to 52.

The difference of 9x and 8x is 17.

The difference of 5c and 4c is 60.

The difference of n and \(\frac{1}{6}\) is \(\frac{1}{2}\).

The difference of f and \(\frac{1}{3}\) is \(\frac{1}{12}\).

The sum of −4n and 5n is −32.

The sum of −9m and 10m is −25.

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Pilar drove from home to school and then to her aunt’s house, a total of 18 miles. The distance from Pilar’s house to school is 7 miles. What is the distance from school to her aunt’s house?

Jeff read a total of 54 pages in his English and Psychology textbooks. He read 41 pages in his English textbook. How many pages did he read in his Psychology textbook?

Pablo’s father is 3 years older than his mother. Pablo’s mother is 42 years old. How old is his father?

Eva’s daughter is 5 years younger than her son. Eva’s son is 12 years old. How old is her daughter?

Allie weighs 8 pounds less than her twin sister Lorrie. Allie weighs 124 pounds. How much does Lorrie weigh?

For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday dinner turkey weighed 16 pounds. How much did the Thanksgiving turkey weigh?

The nurse reported that Tricia’s daughter had gained 4.2 pounds since her last checkup and now weighs 31.6 pounds. How much did Tricia’s daughter weigh at her last checkup?

Connor’s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?

Melissa’s math book cost $22.85 less than her art book cost. Her math book cost $93.75. How much did her art book cost?

Ron’s paycheck this week was $17.43 less than his paycheck last week. His paycheck this week was $103.76. How much was Ron’s paycheck last week?

Everyday Math

Baking Kelsey needs \(\frac{2}{3}\) cup of sugar for the cookie recipe she wants to make. She only has \(\frac{1}{4}\) cup of sugar and will borrow the rest from her neighbor. Let s equal the amount of sugar she will borrow. Solve the equation \(\frac{1}{4}\) + s = \(\frac{2}{3}\) to find the amount of sugar she should ask to borrow.

Construction Miguel wants to drill a hole for a \(\frac{5}{8}\)-inch screw. The screw should be \(\frac{1}{12}\) inch larger than the hole. Let d equal the size of the hole he should drill. Solve the equation d + \(\frac{1}{12}\) = \(\frac{5}{8}\) to see what size the hole should be.

Writing Exercises

Is −18 a solution to the equation 3x = 16 − 5x ? How do you know?

Write a word sentence that translates the equation y − 18 = 41 and then make up an application that uses this equation in its solution.

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Recommended articles

The LibreTexts libraries are Powered by MindTouch®and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Have questions or comments? For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org.